Math 129 Final Exam Fall 2020 11 18 2020 online submitted on Gradescope Directions Please read You must show all your work and justify your methods to obtain full credit Name any theorems that you use Clearly indicate your nal answers Simplify your answers to a reasonable degree You may use your personal notes your book and a calculator during the test You can use your cell phone only at the end of the test to take pictures of your paper and submit Collaboration with other students external people is strictly forbidden Your camera should be on and your microphone muted at all times during the test If you experience a technical issue please let your instructor know immediately For any question to your instructor TA please use the chat The test lasts two hours You will have additional 20 minutes to submit on Gradescope Please use the extra time exclusively for submission Late submissions will be accepted only under exceptional circumstances Please indicate the question number part you are working on on your paper Once you are done you should submit one pdf on Gradescope Please match the pages with the question numbers on the outline Remember USC considers cheating to be a serious o ense the minimum penalty is failure for the course In the current circumstances cheating includes communication with your peers via email social media etc Problem 1 Find the following limits if they exist including In case they do not exist explain why a lim x 0 e2x 1 tan 1 x b lim x 0 1 2x x 1 x2 c lim x x cid 16 2 arctan x cid 17 20 pt Problem 2 Evaluate the following integrals cid 90 x sin x dx x2 9 x2 3 2 dx a b c cid 90 cid 90 20 pt sin3 x cos 3x dx Hint cos 3x 4 cos3 x 3 cos x Problem 3 a Determine whether the improper integral cid 90 0 xe 2x dx converges or diverges by carefully evaluating it b Determine whether dx converges or diverges using any valid argument cid 90 81 x15 7 x1 1 x9 x6 27 20 pt Problem 4 The region R is bounded by x yey and y 1 rotating R about the y axis e x Find the volume obtained by 20 pt Problem 5 A planter in the shape of a trapezoidal prism see picture below is lled with water The bottom of the planter is a 1m by 3m rectangle and the front and back faces are isosceles trapezoids Both trapezoids have bases of 1m and 2m and a height of 1m Recall the density of water is 1000 kg m3 and g 9 8 m s2 a SET UP but DO NOT EVALUATE an integral representing the hydrostatic force on the front face of the planter b SET UP but DO NOT EVALUATE an integral representing the work required to pump all the water out of the top of the planter 20 pt Problem 6 Consider the power series cid 88 n 1 n rn nn x 3 n where r 0 20 pt a Find its radius of convergence as a function of r Hint limn 1 1 n n e b Find a value of r 0 such that the series convergent at x 4 and divergent at x 0 Problem 7 Determine whether the given numerical series are convergent or divergent Justify your answer in each case ln5 n2 n n a b cid 88 n 1 cid 88 n 1 1 n tan 1 n 20 pt Problem 8 A wire is bent into a circle of radius R and a current is run through it It exerts a force on a positively charged particle located distance x above the center of the circle given by where q 0 is a constant f x qx x2 R2 3 2 a Find the Taylor series of f x centered at a 0 b Find the Taylor polynomial of degree 3 for the above series c Use the above Taylor polynomial to estimate f R 2 d Estimate the error in the above approximation 20 pt